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Added some relevant bits to connected limit, fiber product, and pushout. I wanted to record the result at connected limit that functors preserve connected limits iff they preserve wide pullbacks, which may be a slightly surprising result if one has never seen it before.
I added a few facts from Bob Pare’s article “Simply connected limits” to connected limit, fiber product, etc., and along the way created L-finite category.
I added to connected limit a proof that pullbacks and equalizers generate all finite connected limits. I’ve seen it asserted that wide pullbacks and equalizers generate all connected limits, but I don’t see immediately how to generalize this proof to the infinite case. Anyone help?
I also moved the lemma “wide pullbacks + terminal object => complete” to the page wide pullback (which is actually fiber product).
I am tempted to split wide pullback off from fiber product; any thoughts? (Actually, I personally think fiber product ought to redirect to pullback, but I’ll defer on that one.)
Hmm, now I think I see a way to approach the infinite case. Wide pullbacks imply cofiltered limits, and we should be able to construct an arbitrary connected limit as a cofiltered limit of finite connected limits, or at least of L-finite (e.g. finitely generated) connected limits. That makes intuitive sense to me, but it seems like it might be tricky to prove.
I agree with splitting off wide pullback. I also support redirecting fiber product.
Okay, since no one else expressed an opinion for a week, I turned “fiber product” into wide pullback and redirected “fiber product” to pullback.
But the cache bug is operative at fiber product.
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